Question: We know that $ \dfrac{\ln(n)}{n^2}>\dfrac{1}{{{n}^{2}}}>0$ for any $n\ge 3$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=3}^{\infty }~\dfrac{\ln(n)}{n^2}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Explanation: $\sum\limits_{n=1}^{\infty }{\frac{1}{{{n}^{2}}}}$ is a $p$ -series with $p=2$, so it converges. Because our given series is term-by-term greater than a convergent series, the direct comparison test does not apply. So the direct comparison test is inconclusive.